A Square and a Cube — Class 8 Mathematics NCERT Solutions (Free)

Free step-by-step NCERT solutions for Class 8 Mathematics chapter "A Square and a Cube" — 8 important questions with detailed answers for CBSE board exam preparation.

TL;DR: Free step-by-step NCERT solutions for Class 8 Mathematics chapter "A Square and a Cube" — 8 important questions with detailed answers for CBSE board e…

By Syllab.in · Updated Jun 14, 2026

Key Questions Covered:

Find the square root of 256.
Which of the following is a perfect square? 64, 100, 128, 150
Find the cube root of 1000.
Is 216 a perfect cube? Justify your answer.
Find √(144/225).
Find the smallest number by which 75 must be multiplied to make it a perfect …
+ 2 more questions in the full chapter

Solutions Summary:

Question	Status
Find the square root of 256.	✓ Solved
Which of the following is a perfect square? 64, 100, 128,…	✓ Solved
Find the cube root of 1000.	✓ Solved
Is 216 a perfect cube? Justify your answer.	✓ Solved
Find √(144/225).	✓ Solved
Find the smallest number by which 75 must be multiplied t…	✓ Solved

Showing 6 of 8 questions

Q1: Find the square root of 256.

We need to find a number that when multiplied by itself gives 256. Step 1: Let √256 = x, so x² = 256 Step 2: We can find this by prime factorization. 256 = 2 × 128 = 2 × 2 × 64 = 2 × 2 × 2 × 32 = 2 × 2 × 2 × 2 × 16 = 2 × 2 × 2 × 2 × 2 × 8 = 2 × 2 × 2 × 2 × 2 × 2 × 4 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁸ Step 3: √256 = √(2⁸) = 2⁴ = 16 Step 4: Verification: 16 × 16 = 256 ✓ Answer: √256 = 16

Q2: Which of the following is a perfect square? 64, 100, 128, 150

A perfect square is a number that can be expressed as the square of a whole number. Step 1: Check 64 64 = 8 × 8 = 8² ✓ Perfect square Step 2: Check 100 100 = 10 × 10 = 10² ✓ Perfect square Step 3: Check 128 128 = 2 × 64 = 2 × 8² This is not a perfect square because 128 = 2⁷, and the exponent is odd. ✗ Not a perfect square Step 4: Check 150 150 = 2 × 75 = 2 × 3 × 25 = 2 × 3 × 5² Since the exponents of 2 and 3 are odd, this is not a perfect square. ✗ Not a perfect square Answer: 64 and 100 ar...

Q3: Find the cube root of 1000.

We need to find a number that when multiplied by itself three times gives 1000. Step 1: Let ∛1000 = x, so x³ = 1000 Step 2: Prime factorization of 1000 1000 = 10 × 100 = 10 × 10 × 10 = 10³ or 1000 = 2 × 500 = 2 × 2 × 250 = 2 × 2 × 2 × 125 = 2³ × 125 = 2³ × 5³ Step 3: ∛1000 = ∛(10³) = 10 or ∛1000 = ∛(2³ × 5³) = ∛(2 × 5)³ = ∛(10³) = 10 Step 4: Verification: 10 × 10 × 10 = 1000 ✓ Answer: ∛1000 = 10

Q4: Is 216 a perfect cube? Justify your answer.

Step 1: Find the prime factorization of 216 216 = 2 × 108 = 2 × 2 × 54 = 2 × 2 × 2 × 27 = 2³ × 27 = 2³ × 3³ Step 2: Express as a product of prime factors 216 = 2³ × 3³ = (2 × 3)³ = 6³ Step 3: Check the exponents For a number to be a perfect cube, all prime factors must have exponents divisible by 3. Here, both 2 and 3 have exponent 3, which is divisible by 3. Step 4: Verification: 6 × 6 × 6 = 36 × 6 = 216 ✓ Answer: Yes, 216 is a perfect cube because 216 = 6³

Q5: Find √(144/225).

We need to find the square root of a fraction. Step 1: Use the property √(a/b) = √a/√b √(144/225) = √144/√225 Step 2: Find √144 144 = 12 × 12 = 12² √144 = 12 Step 3: Find √225 225 = 15 × 15 = 15² √225 = 15 Step 4: Divide √(144/225) = 12/15 Step 5: Simplify by finding GCD of 12 and 15 GCD(12, 15) = 3 12/15 = (12÷3)/(15÷3) = 4/5 Answer: √(144/225) = 4/5 or 0.8

Q6: Find the smallest number by which 75 must be multiplied to make it a perfect square.

Step 1: Find the prime factorization of 75 75 = 3 × 25 = 3 × 5² Step 2: Analyze the exponents For a perfect square, all prime factors must have even exponents. - Prime factor 3 has exponent 1 (odd) - Prime factor 5 has exponent 2 (even) Step 3: Determine what to multiply To make the exponent of 3 even, we need to multiply by 3 (to make it 3²). The exponent of 5 is already even. Step 4: The number to multiply is 3 75 × 3 = 225 Step 5: Verify that 225 is a perfect square 225 = 3² × 5² = (3 × 5...

Showing 6 of 8 questions. Visit the full page for complete solutions.